The main result of this paper concerns Theorem REF . Going back to the statement, it is important to note that it is sharp when \(k=d=2\) . This is because, by design, every \(K_{2,2}\) -free bipartite graph has \(VC\) -dimension at most 2, and so the well-known examples of such graphs with \(\Omega \left(n^{3/2}\right)\)  edges automatically serve as constructions which match the upper bound from Theorem REF . However, it turns out this phenomenon does not extend for larger \(k\) . For example, already when \(k=3\) , it is not difficult to check that the dense \(K_{3,3}\) -free graph from Brown's construction [1]} has VC-dimension equal to 4, so it does not anymore provide a matching lower bound. This suggests that a potential improvement of Theorem REF  might be possible when \(k \ge d \ge 3\) .
